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3101Two-Step Forecast from Today’s VarianceFor a GARCH(1,1) process with \omega= 1 10 , = 1 5 , = 3 5 , suppose you already know the one-step-ahead conditional variance h t+1 =2. Compute E t[h t+2 ] and E t[h t+3 ].统计中等derivation未尝试面试订阅3103Two-Day Ahead Variance MeanFor a GARCH(1,1) process with \omega=1, = 1 10 , = 4 5 , suppose you already know the one-step-ahead conditional variance h t+1 =5. Compute E t[h t+2 ] and E t[h t+3 ].统计中等derivation未尝试面试订阅3106Half-Life When Alpha Plus Beta Equals 0.8In a GARCH-style volatility recursion, a deviation from long-run variance decays approximately by the factor = + = 4 5 each step. What is the half-life of the deviation?统计中等derivation未尝试面试订阅3111Does This GARCH Have a Finite Long-Run Variance?For a GARCH(1,1) model with \omega= 1 5 , = 1 4 , and = 3 4 , decide whether the model has a finite unconditional variance. If it does, compute it.统计中等derivation未尝试面试订阅3116Posterior Buy Probability After 7 Buys and 3 SellsA Bernoulli success probability p has prior Beta (2,3). After observing 7 successes and 3 failures, what is the posterior mean of p?统计简单derivation未尝试面试订阅3123Probability the Next Fill Is a SuccessA Bernoulli success probability has prior Beta (3,5). After observing 6 successes and 2 failures, compute the requested posterior predictive probability for the next 1 trial(s).统计中等derivation未尝试面试订阅3126Posterior Mean Arrival Rate After 12 Events in Two HoursA Poisson rate has prior Gamma (3,1) in shape-rate form. After observing 12 events over 2 hour(s), what is the posterior mean of ?统计中等derivation未尝试面试订阅3128Next-Hour Predictive Mean from Gamma-PoissonA Poisson rate has prior Gamma (4,1) in shape-rate form. After observing 5 events over 1 hour(s), what is the posterior predictive mean number of events in the next 1 hour(s)?统计中等derivation未尝试面试订阅3130Predictive Mean Trade Count in the Next Half HourA Poisson rate has prior Gamma (5,2) in shape-rate form. After observing 10 events over 4 hour(s), what is the posterior predictive mean number of events in the next 1 2 hour(s)?统计中等derivation未尝试面试订阅3131Posterior Mean of a Latent Fair ValueSuppose \sim N(0,4) and conditional on , you observe n=1 independent measurements with known variance 2=2 and sample mean x=3. Compute the posterior mean and posterior variance of .统计中等derivation未尝试面试订阅3133Posterior Mean from Three Noisy MeasurementsSuppose \sim N(-1,16) and conditional on , you observe n=3 independent measurements with known variance 2=1 and sample mean x=2. Compute the posterior mean and posterior variance of .统计中等derivation未尝试面试订阅3136Posterior Predictive Category ProbabilitiesA categorical distribution over three states has prior Dirichlet (2,3,5). After observing counts [4, 1, 5], what is the posterior predictive probability that the next observation is in category 1?统计中等derivation未尝试面试订阅3141Posterior After HHTTA Bernoulli success probability has prior Beta (1,1). Data arrive sequentially with outcome string `HHTT` where each non-`T` symbol is a success and each `T` is a failure. What is the posterior after processing the full sequence, and what is the posterior predictive probability of success on the next trial?统计中等derivation未尝试面试订阅3146Daily Count Posterior After Three DaysA Poisson rate has prior Gamma (2,1) in shape-rate form. You observe sequential unit-time counts [3, 4, 2]. What is the final posterior for , and what is its posterior mean after the last update?统计中等derivation未尝试面试订阅3147Posterior Rate After Four Trading SessionsA Poisson rate has prior Gamma (1,2) in shape-rate form. You observe sequential unit-time counts [1, 0, 2, 3]. What is the final posterior for , and what is its posterior mean after the last update?统计中等derivation未尝试面试订阅3148Online Call-Rate UpdateA Poisson rate has prior Gamma (4,1) in shape-rate form. You observe sequential unit-time counts [5, 6]. What is the final posterior for , and what is its posterior mean after the last update?统计中等derivation未尝试面试订阅3150Posterior Mean After Five Intraday BucketsA Poisson rate has prior Gamma (2,2) in shape-rate form. You observe sequential unit-time counts [0, 1, 2, 1, 3]. What is the final posterior for , and what is its posterior mean after the last update?统计中等derivation未尝试面试订阅3151Posterior Mean After Two Streaming MeasurementsA latent mean has prior N(0,4). Observations arrive one by one with known noise variance 2=1 and realized values [2, 4]. After processing the full stream sequentially, what are the final posterior mean and variance?统计中等derivation未尝试面试订阅3152Posterior Mean After Three Sequential Forecast ErrorsA latent mean has prior N(5,9). Observations arrive one by one with known noise variance 2=4 and realized values [6, 5, 3]. After processing the full stream sequentially, what are the final posterior mean and variance?统计中等derivation未尝试面试订阅3153Online Update of a Latent DriftA latent mean has prior N(-1,16). Observations arrive one by one with known noise variance 2=1 and realized values [0, 2]. After processing the full stream sequentially, what are the final posterior mean and variance?统计中等derivation未尝试面试订阅